0.6 Regularity, moments, and wavelet system design  (Page 7/13)

Based on the above definition, one observes $\phi$ has to be a constant if $\alpha >1$ . Hence, this is not very useful for determining regularity of order $\alpha >1$ . However, using the above definition, Hölder regularity of any order $r>0$ is defined as follows:

Definition 2 (Hölder regularity) A function $\phi :\text{ℝ}\to \text{ℂ}$ is regular of order $r=P+\alpha \text{(0<α≤1)}$ if $\phi \in C^P$ and its Pth derivative is Hölder continuous of order $\alpha$

Definition 3 (Sobolev regularity) $\text{Let}\phi :\text{ℝ}\to \text{ℂ,}\text{then}\phi \text{is said to belong to the Sobolev space of order s}(\phi \in H_s)\text{if}$

Notice that, although Sobolev regularity does not give the explicit order of differentiability, it does yield lower and upper bounds on $r$ , the Hölder regularity, and hence the differentiability of $\phi$ if $\phi \in L^2$ . This can be seen from the following inclusions:

A very interesting and important result by Volkmer [link] and by Eirola [link] gives an exact asymptotic formula for the Hölder regularity index (exponent) ofthe Daubechies scaling function.

Theorem 24 The limit of the Hölder regularity index of a Daubechies scaling function as the length of the scaling filter goes to infinity is [link]

This result, which was also proven by A. Cohen and J. P. Conze, together with empirical calculations for shorter lengths, gives a good picture ofthe smoothness of Daubechies scaling functions. This is illustrated in [link] where the Hölder index is plotted versus scaling filter length for both the maximally smooth case and the Daubechies case.

The question of the behavior of maximally smooth scaling functions was empirically addressed by Lang and Heller in [link] . They use an algorithm by Rioul to calculate the Hölder smoothness of scalingfunctions that have been designed to have maximum Hölder smoothness and the results are shown in [link] together with the smoothness of the Daubechies scaling functions as functions of the lengthof the scaling filter. For the longer lengths, it is possible to design systems that give a scaling function over twice as smooth as with aDaubechies design. In most applications, however, the greater Hölder smoothness is probably not important.

[link] shows the number of zero moments (zeros at $\omega =\pi$ ) as a function of the number of scaling function coefficients for both the maximally smooth and Daubechies designs.

One case from this figure is for $N=26$ where the Daubechies smoothness is $S_H=4.005$ and the maximum smoothness is $S_H=5.06$ . The maximally smooth scaling function has one more continuous derivative than theDaubechies scaling function.

Recent work by Heller and Wells [link] , [link] gives a better connection of properties of the scaling coefficients and the smoothness of thescaling function and wavelets. This is done both for the scale factor or multiplicity of $M=2$ and for general integer $M$ .

The usual definition of smoothness in terms of differentiability may not be the best measure for certain signal processing applications. If thesignal is given as a sequence of numbers, not as a function of a continuous variable, what does smoothness mean? Perhaps the use of the variation of a signal may be a useful alternative [link] , [link] , [link] .